Integrand size = 24, antiderivative size = 83 \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\arctan (a x)^{3/2}}{12 a^3 c^3}+\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{64 a^3 c^3}-\frac {\sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{32 a^3 c^3} \]
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Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5090, 4491, 3377, 3386, 3432} \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{64 a^3 c^3}+\frac {\arctan (a x)^{3/2}}{12 a^3 c^3}-\frac {\sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{32 a^3 c^3} \]
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Rule 3377
Rule 3386
Rule 3432
Rule 4491
Rule 5090
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sqrt {x} \cos ^2(x) \sin ^2(x) \, dx,x,\arctan (a x)\right )}{a^3 c^3} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\sqrt {x}}{8}-\frac {1}{8} \sqrt {x} \cos (4 x)\right ) \, dx,x,\arctan (a x)\right )}{a^3 c^3} \\ & = \frac {\arctan (a x)^{3/2}}{12 a^3 c^3}-\frac {\text {Subst}\left (\int \sqrt {x} \cos (4 x) \, dx,x,\arctan (a x)\right )}{8 a^3 c^3} \\ & = \frac {\arctan (a x)^{3/2}}{12 a^3 c^3}-\frac {\sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{32 a^3 c^3}+\frac {\text {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{64 a^3 c^3} \\ & = \frac {\arctan (a x)^{3/2}}{12 a^3 c^3}-\frac {\sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{32 a^3 c^3}+\frac {\text {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{32 a^3 c^3} \\ & = \frac {\arctan (a x)^{3/2}}{12 a^3 c^3}+\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{64 a^3 c^3}-\frac {\sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{32 a^3 c^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.70 \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {32 \arctan (a x) \left (3 a x \left (-1+a^2 x^2\right )+2 \left (1+a^2 x^2\right )^2 \arctan (a x)\right )-3 \left (1+a^2 x^2\right )^2 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )-3 \left (1+a^2 x^2\right )^2 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )}{768 a^3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}} \]
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Time = 5.65 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {3 \,\operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }+32 \arctan \left (a x \right )^{2}-12 \sin \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )}{384 c^{3} a^{3} \sqrt {\arctan \left (a x \right )}}\) | \(66\) |
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Exception generated. \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{2} \sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
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Exception generated. \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x^{2} \sqrt {\arctan \left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {x^2\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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